Transcription factor search for a DNA promoter in a three-state model

To ensure fast gene activation, transcription factors (TFs) use a mechanism known as facilitated diffusion to find their DNA promoter site. Here we analyze such a process where a TF alternates between three- and one-dimensional diffusion. In the latter (TF bound to the DNA), the TF further switches between a fast translocation state dominated by interaction with the DNA backbone, and a slow examination state where interaction with DNA base pairs (bp) is predominant. We derive a formula for the mean search time, and show that it is faster and less sensitive to the binding-energy fluctuations as compared to the case with a single sliding state. We find that for an optimal search, the time spent bound to the DNA is larger compared to the three-dimensional time, in agreement with recent experimental data.